Richard Wolfenden
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The organizations/institutes from the scholar's latest article
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The organizations / institutes from the scholar's other articles in the past ten years
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Richard Wolfenden has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
Charles A. Lewis
Department of Biochemistry and Biophysics, University of North Carolina, 120 Mason Farm Road, Chapel Hill, North Carolina 27154, United States
12 shared publications- Randy B. Stockbridge
University of Michigan
5 shared publications Yang Yuan
Department of Biochemistry and Biophysics, University of North Carolina, Chapel Hill, NC 27599
5 shared publicationsChetan Lad
Centre for Chemical Biology, Krebs Institute for Biomolecular Science, Department of Chemistry, University of Sheffield, Sheffield S3 7HF, United Kingdom
2 shared publicationsGottfried K. Schroeder
2 shared publications- Nicholas H. Williams
Department of Chemistry, University of Sheffield, Sheffield S3 7HF, UK
2 shared publications
Latest Publications
- 27 February 2023
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 62 (5) , 1026-1031
Aldol Cleavage in Water and the Power of Citrate Lyase as a Catalyst
Citrate lyase allows Klebsiella aerogenes to grow anaerobically on citrate as the sole carbon source. Arrhenius analysis of experiments at high temperatures indicates that citrate is cleaved nonenzymatically to acetate and oxaloacetate with a t1/2 of 6.9 million years in neutral solution at 25 °C, while malate cleavage occurs even more slowly (t1/2 = 280 million years). However, t1/2 is only 10 days for the nonenzymatic cleavage of 4-hydroxy-2-ketoglutarate, indicating that the introduction of an α-keto group enhances the rate of aldol cleavage of malate by a factor of 1010. The aldol cleavages of citrate and malate, like the decarboxylation of malonate (t1/2 = 180 years), are associated with a near-zero entropy of activation, and their extreme differences in rate reflect differences between their heats of activation. Citrate lyase enhances the rate of substrate cleavage 6 × 1015-fold, comparable in magnitude with the rate enhancement produced by OMP decarboxylase, although these enzymes are strikingly different in their mechanisms of action.
- 27 February 2023
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 62 (5) , 1026-1031
Aldol Cleavage in Water and the Power of Citrate Lyase as a Catalyst
Citrate lyase allows Klebsiella aerogenes to grow anaerobically on citrate as the sole carbon source. Arrhenius analysis of experiments at high temperatures indicates that citrate is cleaved nonenzymatically to acetate and oxaloacetate with a t1/2 of 6.9 million years in neutral solution at 25 °C, while malate cleavage occurs even more slowly (t1/2 = 280 million years). However, t1/2 is only 10 days for the nonenzymatic cleavage of 4-hydroxy-2-ketoglutarate, indicating that the introduction of an α-keto group enhances the rate of aldol cleavage of malate by a factor of 1010. The aldol cleavages of citrate and malate, like the decarboxylation of malonate (t1/2 = 180 years), are associated with a near-zero entropy of activation, and their extreme differences in rate reflect differences between their heats of activation. Citrate lyase enhances the rate of substrate cleavage 6 × 1015-fold, comparable in magnitude with the rate enhancement produced by OMP decarboxylase, although these enzymes are strikingly different in their mechanisms of action.
- Open Access
The Burden Borne by Protein Methyltransferases: Rates and Equilibria of Nonenzymatic Methylation of Amino Acid Side-Chains by SAM in Water
SAM is a powerful methylating agent, with a methyl group transfer potential matching the phosphoryl group transfer potential of ATP. SAM-dependent N-methyltransferases have evolved to catalyze the modification of specific lysine residues in histones and transcription factors, in addition to generating epinephrine, N-methylnicotinamide, and a quaternary amine (betaine) that is used to maintain osmotic pressure in plants and halophilic bacteria. To assess the catalytic power of these enzymes and their potential susceptibility to transition state and multisubstrate analogue inhibitors, we determined the rates and positions of equilibrium of methyl transfer from the trimethylsulfonium ion to model amines in the absence of a catalyst. Unlike the methyl group transfer potential of SAM, which becomes more negative with increasing pH throughout the normal pH range, equilibrium constants for the hydrolytic demethylation of secondary, tertiary and quaternary amines are found to be insensitive to changing pH and resemble to each other in magnitude, with an average ΔG value of ∼ -0.7 kcal/mol at pH 7. Thus, each of the three steps in the mono- di- and trimethylation of lysine by SAM is accompanied by a free energy change of -7.5 kcal/mol in neutral solution. Arrhenius analysis of the uncatalyzed reactions shows that the unprotonated form of glycine attacks the trimethylsulfonium ion (TMS+ + ) with a second order rates constant of 1.8 × 10 −7 M -1 s -1 at 25 °C (ΔH ‡ = 22 kcal/mol and TΔS ‡ = -6 kcal/mol). Comparable values are observed for the methylation of secondary and tertiary amines, with k 25 = 1.1 × 10 −7 M -1 s -1 for sarcosine and 4.3 × 10 −8 M -1 s -1 for dimethylglycine. The nonenzymatic methylation of imidazole and methionine by TMS+ + , benchmarks for the methylation of histidine and methionine residues by SETD3, exhibit k 25 values of 3.3 × 10 −9 and 1.2 × 10 −9 M -1 s -1 respectively. Lysine methylation by SAM, although slow under physiological conditions (t 1/2 7 weeks at 25 °C), is accelerated 1.1 × 10 12 -fold at the active site of a SET domain methyltransferase.
Open AccessThe Burden Borne by Protein Methyltransferases: Rates and Equilibria of Nonenzymatic Methylation of Amino Acid Side-Chains by SAM in Water
SAM is a powerful methylating agent, with a methyl group transfer potential matching the phosphoryl group transfer potential of ATP. SAM-dependent N-methyltransferases have evolved to catalyze the modification of specific lysine residues in histones and transcription factors, in addition to generating epinephrine, N-methylnicotinamide, and a quaternary amine (betaine) that is used to maintain osmotic pressure in plants and halophilic bacteria. To assess the catalytic power of these enzymes and their potential susceptibility to transition state and multisubstrate analogue inhibitors, we determined the rates and positions of equilibrium of methyl transfer from the trimethylsulfonium ion to model amines in the absence of a catalyst. Unlike the methyl group transfer potential of SAM, which becomes more negative with increasing pH throughout the normal pH range, equilibrium constants for the hydrolytic demethylation of secondary, tertiary and quaternary amines are found to be insensitive to changing pH and resemble to each other in magnitude, with an average ΔG value of ∼ -0.7 kcal/mol at pH 7. Thus, each of the three steps in the mono- di- and trimethylation of lysine by SAM is accompanied by a free energy change of -7.5 kcal/mol in neutral solution. Arrhenius analysis of the uncatalyzed reactions shows that the unprotonated form of glycine attacks the trimethylsulfonium ion (TMS+ + ) with a second order rates constant of 1.8 × 10 −7 M -1 s -1 at 25 °C (ΔH ‡ = 22 kcal/mol and TΔS ‡ = -6 kcal/mol). Comparable values are observed for the methylation of secondary and tertiary amines, with k 25 = 1.1 × 10 −7 M -1 s -1 for sarcosine and 4.3 × 10 −8 M -1 s -1 for dimethylglycine. The nonenzymatic methylation of imidazole and methionine by TMS+ + , benchmarks for the methylation of histidine and methionine residues by SETD3, exhibit k 25 values of 3.3 × 10 −9 and 1.2 × 10 −9 M -1 s -1 respectively. Lysine methylation by SAM, although slow under physiological conditions (t 1/2 7 weeks at 25 °C), is accelerated 1.1 × 10 12 -fold at the active site of a SET domain methyltransferase.
- 5 March 2021
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 60 (11) , 854-858
The Burden Borne by Protein Methyltransferases: Rates and Equilibria of Non-enzymatic Methylation of Amino Acid Side Chains by SAM in Water
SAM is a powerful methylating agent, with a methyl group transfer potential matching the phosphoryl group transfer potential of ATP. SAM-dependent N-methyltransferases have evolved to catalyze the modification of specific lysine residues in histones and transcription factors, in addition to generating epinephrine, N-methylnicotinamide, and a quaternary amine (betaine) that is used to maintain osmotic pressure in plants and halophilic bacteria. To assess the catalytic power of these enzymes and their potential susceptibility to transition state and multisubstrate analogue inhibitors, we determined the rates and positions of the equilibrium of methyl transfer from the trimethylsulfonium ion to model amines in the absence of a catalyst. Unlike the methyl group transfer potential of SAM, which becomes more negative with an increase in pH throughout the normal pH range, equilibrium constants for the hydrolytic demethylation of secondary, tertiary, and quaternary amines are found to be insensitive to a change in pH and resemble each other in magnitude, with an average ΔG value of approximately −0.7 kcal/mol at pH 7. Thus, each of the three steps in the mono-, di-, and trimethylation of lysine by SAM is accompanied by a change in free energy of −7.5 kcal/mol in a neutral solution. Arrhenius analysis of the uncatalyzed reactions shows that the unprotonated form of glycine attacks the trimethylsulfonium ion (TMS+) with second-order rates constant of 1.8 × 10–7 M–1 s–1 at 25 °C (ΔH⧧ = 22 kcal/mol, and TΔS⧧ = −6 kcal/mol). Comparable values are observed for the methylation of secondary and tertiary amines, with k25 values of 1.1 × 10–7 M–1 s–1 for sarcosine and 4.3 × 10–8 M–1 s–1 for dimethylglycine. The non-enzymatic methylations of imidazole and methionine by TMS+, benchmarks for the methylation of histidine and methionine residues by SETD3, exhibit k25 values of 3.3 × 10–9 and 1.2 × 10–9 M–1 s–1, respectively. Lysine methylation by SAM, although slow under physiological conditions (t1/2 = 7 weeks at 25 °C), is accelerated 1.1 × 1012 -fold at the active site of a SET domain methyltransferase.
- 5 March 2021
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 60 (11) , 854-858
The Burden Borne by Protein Methyltransferases: Rates and Equilibria of Non-enzymatic Methylation of Amino Acid Side Chains by SAM in Water
SAM is a powerful methylating agent, with a methyl group transfer potential matching the phosphoryl group transfer potential of ATP. SAM-dependent N-methyltransferases have evolved to catalyze the modification of specific lysine residues in histones and transcription factors, in addition to generating epinephrine, N-methylnicotinamide, and a quaternary amine (betaine) that is used to maintain osmotic pressure in plants and halophilic bacteria. To assess the catalytic power of these enzymes and their potential susceptibility to transition state and multisubstrate analogue inhibitors, we determined the rates and positions of the equilibrium of methyl transfer from the trimethylsulfonium ion to model amines in the absence of a catalyst. Unlike the methyl group transfer potential of SAM, which becomes more negative with an increase in pH throughout the normal pH range, equilibrium constants for the hydrolytic demethylation of secondary, tertiary, and quaternary amines are found to be insensitive to a change in pH and resemble each other in magnitude, with an average ΔG value of approximately −0.7 kcal/mol at pH 7. Thus, each of the three steps in the mono-, di-, and trimethylation of lysine by SAM is accompanied by a change in free energy of −7.5 kcal/mol in a neutral solution. Arrhenius analysis of the uncatalyzed reactions shows that the unprotonated form of glycine attacks the trimethylsulfonium ion (TMS+) with second-order rates constant of 1.8 × 10–7 M–1 s–1 at 25 °C (ΔH⧧ = 22 kcal/mol, and TΔS⧧ = −6 kcal/mol). Comparable values are observed for the methylation of secondary and tertiary amines, with k25 values of 1.1 × 10–7 M–1 s–1 for sarcosine and 4.3 × 10–8 M–1 s–1 for dimethylglycine. The non-enzymatic methylations of imidazole and methionine by TMS+, benchmarks for the methylation of histidine and methionine residues by SETD3, exhibit k25 values of 3.3 × 10–9 and 1.2 × 10–9 M–1 s–1, respectively. Lysine methylation by SAM, although slow under physiological conditions (t1/2 = 7 weeks at 25 °C), is accelerated 1.1 × 1012 -fold at the active site of a SET domain methyltransferase.
- 28 October 2019
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 58 (52) , 5381-5385
Ether Hydrolysis, Ether Thiolysis, and the Catalytic Power of Etherases in the Disassembly of Lignin
The recycling of much of the carbon in Nature depends on the breakdown of polymers in woody matter, notably cellulose (a polyacetal) and lignin (a polyether). Here, we show that equilibrium favors ether hydrolysis in water, but that the rates of hydrolysis of ethers are too slow to measure in neutral solution except at temperatures approaching the critical point of water. Circumventing that obstacle, glutathione-dependent etherases from white-rot fungi are known to employ the thiolate group of glutathione to attack guaiacyl ethers. The present experiments at elevated temperatures indicate that thioglycolate attacks diethyl ether in water—in the absence of enzymes—with a rate constant of 6 x 10^-11 M^-1 s^-1 at 25 °, and that thiolysis is strongly favored thermodynamically, with a K(v)eq value of 2.5 x 10^6 (ΔG ~ -8.7 kcal/mol). Compared with the rate of nonenzymatic thiolysis, the lignin-degrading etherases LigE and LigF produce 10^15-fold rate enhancements, among the largest that have been observed for an enzyme acting on two substrates. As in other enzymatic displacement reactions, one function of these enzymes' active sites is presumably to remove the substrates from constraints on their reactivity imposed by solvent water.
- 28 October 2019
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 58 (52) , 5381-5385
Ether Hydrolysis, Ether Thiolysis, and the Catalytic Power of Etherases in the Disassembly of Lignin
The recycling of much of the carbon in Nature depends on the breakdown of polymers in woody matter, notably cellulose (a polyacetal) and lignin (a polyether). Here, we show that equilibrium favors ether hydrolysis in water, but that the rates of hydrolysis of ethers are too slow to measure in neutral solution except at temperatures approaching the critical point of water. Circumventing that obstacle, glutathione-dependent etherases from white-rot fungi are known to employ the thiolate group of glutathione to attack guaiacyl ethers. The present experiments at elevated temperatures indicate that thioglycolate attacks diethyl ether in water—in the absence of enzymes—with a rate constant of 6 x 10^-11 M^-1 s^-1 at 25 °, and that thiolysis is strongly favored thermodynamically, with a K(v)eq value of 2.5 x 10^6 (ΔG ~ -8.7 kcal/mol). Compared with the rate of nonenzymatic thiolysis, the lignin-degrading etherases LigE and LigF produce 10^15-fold rate enhancements, among the largest that have been observed for an enzyme acting on two substrates. As in other enzymatic displacement reactions, one function of these enzymes' active sites is presumably to remove the substrates from constraints on their reactivity imposed by solvent water.
- 22 May 2018
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 57 (26) , 3549-3551
Sulfonium Ion Condensation: The Burden Borne by SAM Synthetase
S-Adenosylmethionine (SAM+) serves as the prin-cipal methylating agent in biological systems, but the thermodynamic basis of its reactivity does not appear to have been established. Here, we show that methionine, methanol and H+ combine to form S-methylmethionine (SMM+) with a temperature-independent equilibrium constant of 9.9 M-2. The corresponding group transfer potential of SMM+, i. e. its free energy of hydrolysis at pH 7, is -8.2 kcal/mol. The “energy-rich” nature of sulfonium ions is related to the extreme acidity (pKa -5.4) of the S-protonated thioether produced by sulfonium hydrolysis, and the massive negative free energy of deprotonation of that species in neutral solution (-16.7 kcal/mol). At pH 7, SAM synthetase requires the free energy released by cleavage of two bonds of ATP to reverse that process.
- 22 May 2018
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 57 (26) , 3549-3551
Sulfonium Ion Condensation: The Burden Borne by SAM Synthetase
S-Adenosylmethionine (SAM+) serves as the prin-cipal methylating agent in biological systems, but the thermodynamic basis of its reactivity does not appear to have been established. Here, we show that methionine, methanol and H+ combine to form S-methylmethionine (SMM+) with a temperature-independent equilibrium constant of 9.9 M-2. The corresponding group transfer potential of SMM+, i. e. its free energy of hydrolysis at pH 7, is -8.2 kcal/mol. The “energy-rich” nature of sulfonium ions is related to the extreme acidity (pKa -5.4) of the S-protonated thioether produced by sulfonium hydrolysis, and the massive negative free energy of deprotonation of that species in neutral solution (-16.7 kcal/mol). At pH 7, SAM synthetase requires the free energy released by cleavage of two bonds of ATP to reverse that process.
- 6 March 2017
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 56 (10) , 1498-1503
Three Pyrimidine Decarboxylations in the Absence of a Catalyst
The epigenetic modification of DNA by 5-methylation of cytosine residues can be reversed by the action of the TET family of dioxygenases that oxidize the methyl group to produce 5-carboxycytosine (5caC), which can be converted to cytosine in a final decarboxylation step. Likewise, 5-carboxyuracil (5caU) is decarboxylated to uracil in the last step in pyrimidine salvage. In view of the extreme difficulty of decarboxylating derivatives of orotic acid (6caU), it seemed desirable to establish the rates of decarboxylation of 5caC and 5caU in the absence of a catalyst. Arrhenius analysis of experiments performed at elevated temperatures indicates that 5caU decomposes with a rate constant of 1.1 × 10–9 s–1 (ΔH⧧ = 25 kcal/mol) in a neutral solution at 25 °C. The decomposition of 5caC is somewhat slower (k25 = 5.0 × 10–11 s–1; ΔH⧧ = 27 kcal/mol) and leads to the initial accumulation of cytosine as an intermediate, followed by the relatively rapid deamination of cytosine (k25 = 1.9 × 10–10 s–1; ΔH⧧ = 23.4 kcal/mol). Both 5caC and 5caU are decarboxylated many orders of magnitude more rapidly than 6caU is (k25 = 1.3 × 10–17 s–1). Ab initio simulations indicate that in all three cases, the favored route of spontaneous decarboxylation in water involves direct elimination of CO2 with the assistance of an explicit water molecule.
- 6 March 2017
- Published by American Chemical Society (ACS) in Biochemistry
- Vol. 56 (10) , 1498-1503
Three Pyrimidine Decarboxylations in the Absence of a Catalyst
The epigenetic modification of DNA by 5-methylation of cytosine residues can be reversed by the action of the TET family of dioxygenases that oxidize the methyl group to produce 5-carboxycytosine (5caC), which can be converted to cytosine in a final decarboxylation step. Likewise, 5-carboxyuracil (5caU) is decarboxylated to uracil in the last step in pyrimidine salvage. In view of the extreme difficulty of decarboxylating derivatives of orotic acid (6caU), it seemed desirable to establish the rates of decarboxylation of 5caC and 5caU in the absence of a catalyst. Arrhenius analysis of experiments performed at elevated temperatures indicates that 5caU decomposes with a rate constant of 1.1 × 10–9 s–1 (ΔH⧧ = 25 kcal/mol) in a neutral solution at 25 °C. The decomposition of 5caC is somewhat slower (k25 = 5.0 × 10–11 s–1; ΔH⧧ = 27 kcal/mol) and leads to the initial accumulation of cytosine as an intermediate, followed by the relatively rapid deamination of cytosine (k25 = 1.9 × 10–10 s–1; ΔH⧧ = 23.4 kcal/mol). Both 5caC and 5caU are decarboxylated many orders of magnitude more rapidly than 6caU is (k25 = 1.3 × 10–17 s–1). Ab initio simulations indicate that in all three cases, the favored route of spontaneous decarboxylation in water involves direct elimination of CO2 with the assistance of an explicit water molecule.
- 5 July 2016
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 113 (29) , 8194-8199
Cytosine deamination and the precipitous decline of spontaneous mutation during Earth's history
Significance: Cytosine deamination appears to be largely responsible for spontaneous mutations in the modern world. Because of its sensitivity to temperature (Q 10 = 4), that reaction would have furnished a mechanism for rapid evolution on a warm earth. As the temperature fell from 100° to 25 °C, the rate of cytosine-based mutation would have fallen by a factor of more than 4,000, with a corresponding increase in the stability of genetic information. Other potentially mutagenic events are known to be even more sensitive to temperature, and would presumably have led to an even steeper decline in the rate of spontaneous mutation as the earth cooled.
- 5 July 2016
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 113 (29) , 8194-8199
Cytosine deamination and the precipitous decline of spontaneous mutation during Earth's history
Significance: Cytosine deamination appears to be largely responsible for spontaneous mutations in the modern world. Because of its sensitivity to temperature (Q 10 = 4), that reaction would have furnished a mechanism for rapid evolution on a warm earth. As the temperature fell from 100° to 25 °C, the rate of cytosine-based mutation would have fallen by a factor of more than 4,000, with a corresponding increase in the stability of genetic information. Other potentially mutagenic events are known to be even more sensitive to temperature, and would presumably have led to an even steeper decline in the rate of spontaneous mutation as the earth cooled.
- 16 June 2015
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 112 (24) , 7484-7488
Temperature dependence of amino acid hydrophobicities
Significance: Systematic relationships have long been recognized between the hydrophobicities of amino acids and ( i ) their tendencies to be located at the exposed surfaces of globular and membrane proteins and ( ii ) the composition of their triplets in the genetic code. Here, we show that the same coding relationships are compatible with the high temperatures at which life is widely believed to have originated. An accompanying paper reports that these two properties appear to be encoded separately by bases in the acceptor stem and the anticodon of tRNA.
- 16 June 2015
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 112 (24) , 7484-7488
Temperature dependence of amino acid hydrophobicities
Significance: Systematic relationships have long been recognized between the hydrophobicities of amino acids and ( i ) their tendencies to be located at the exposed surfaces of globular and membrane proteins and ( ii ) the composition of their triplets in the genetic code. Here, we show that the same coding relationships are compatible with the high temperatures at which life is widely believed to have originated. An accompanying paper reports that these two properties appear to be encoded separately by bases in the acceptor stem and the anticodon of tRNA.
- 16 June 2015
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 112 (24) , 7489-7494
tRNA acceptor stem and anticodon bases form independent codes related to protein folding
Significance: The universal genetic code is the earliest point to which we can trace biological inheritance. Earlier work hinted at a relationship between the codon bases and the physical properties of the 20 amino acids that dictate the 3D conformations of proteins in solution. Here, we show that acceptor stems and anticodons, which are at opposite ends of the tRNA molecule, code, respectively, for size and polarity. These two distinct properties of the amino acid side-chains jointly determine their preferred locations in folded proteins. The early appearance of an acceptor stem code based on size, β-branching, and carboxylate groups might have favored the appearance of antiparallel peptides that have been suggested to have a special affinity for RNA.
- 16 June 2015
- Published by National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 112 (24) , 7489-7494
tRNA acceptor stem and anticodon bases form independent codes related to protein folding
Significance: The universal genetic code is the earliest point to which we can trace biological inheritance. Earlier work hinted at a relationship between the codon bases and the physical properties of the 20 amino acids that dictate the 3D conformations of proteins in solution. Here, we show that acceptor stems and anticodons, which are at opposite ends of the tRNA molecule, code, respectively, for size and polarity. These two distinct properties of the amino acid side-chains jointly determine their preferred locations in folded proteins. The early appearance of an acceptor stem code based on size, β-branching, and carboxylate groups might have favored the appearance of antiparallel peptides that have been suggested to have a special affinity for RNA.
- Open Access
- 1 October 2014
- Published by Elsevier in Journal of Biological Chemistry
- Vol. 289 (44) , 30198-30204
Massive Thermal Acceleration of the Emergence of Primordial Chemistry, the Incidence of Spontaneous Mutation, and the Evolution of Enzymes
- 1 October 2014
- Published by Elsevier in Journal of Biological Chemistry
- Vol. 289 (44) , 30198-30204
Open AccessMassive Thermal Acceleration of the Emergence of Primordial Chemistry, the Incidence of Spontaneous Mutation, and the Evolution of Enzymes